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  1. ; (, Proceedings of 36th International Symposium on Computational Geometry (SoCG 2020))
    The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ=(δ₁,… ,δ_k) of k distances, a (k+1)-tuple (p₁,… ,p_{k+1}) of distinct points in ℝ^d is called a (k,δ)-chain if ‖p_j-p_{j+1}‖ = δ_j for every 1 ≤ j ≤ k. What is the maximum number C_k^d(n) of (k,δ)-chains in a set of n points in ℝ^d, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3. 
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